Optical process control measurements can often be regarded as methods for measuring parameters of a pattern. For example, a pattern can be a periodic one-dimensional grating of lines on the surface of a wafer, and the parameters to measure can be line width, line spacing and depth of the grating. To measure these parameters, an optical response of the pattern is measured. For example, reflectance as a function of wavelength can be measured. Typically, the optical response will depend on the parameter (or parameters) of interest in a complicated way such that direct parameter extraction from measured data is impractical. Instead, a mathematical model can be constructed for the pattern, having the parameters of interest as variables. Within the model, a modeled optical response is calculated corresponding to the measured optical response. The parameters of interest are then determined by adjusting the variables to fit the modeled response to the measured response.
A commonly-employed modeling approach for grating diffraction is known as the rigorous coupled wave analysis (RCWA). The RCWA is based on retaining a finite number of diffraction orders in the calculation, thereby allowing Maxwell's equations for the grating diffraction problem to be expressed as a system of matrix equations. The RCWA is well known and is described in many references, including Moharam et al., Journal of the Optical Society of America (JOSA), A12, n5, pp1068-1076, 1995, and Moharam et al., JOSA, A12, n5, pp1077-1086, 1995.
Since it is usually necessary to run a large number of cases to fit a modeled grating response to a measured grating response, methods of reducing RCWA calculation time are of special interest, and many approaches for reducing RCWA calculation time have been described. For example, in the second Moharam paper cited above, a layer by layer solution approach is described, where the system of matrix equations is basically solved one interface at a time, as opposed to solving for all interfaces simultaneously. This layer by layer approach can dramatically reduce the computation time compared to the simultaneous approach.
In this Moharam paper, it is also noted that in cases where it is not necessary to calculate both reflectance (R) and transmittance (T), the calculations can be modified to provide only the desired quantity and to significantly reduce computation time. This is in contrast to the more familiar case of R and T calculations relating to planar multi-layer stacks, where calculating only R or only T is not significantly faster than calculating both R and T.
Other approaches have also been considered for reducing RCWA calculation time. In some cases, it is possible to exploit symmetry to reduce calculation time.
For example, U.S. Pat. No. 6,898,537 considers calculations performed for modeling normal incidence illumination. At normal incidence, the positive and negative diffraction orders are not independent, so the RCWA can be reformulated to reduce matrix size, thereby reducing calculation time.
Another approach which has been considered for reducing RCWA calculation time is caching or storing intermediate results, such that recalculation of the cached intermediate results is avoided. Caching in connection with RCWA calculations is considered in U.S. Pat. No. 6,891,626, U.S. Pat. No. 6,952,271, U.S. Pat. No. 7,031,894, and U.S. Pat. No. 7,099,005. FIG. 1 shows a representative example of a conventional RCWA method along the lines of these references. In this example, step 102 is to measure a spectral reflectance (i.e., Rmeas(λ)). Step 104 is to select a combination of parameter values for the grating. Step 106 is to calculate a modeled spectral reflectance (i.e., Rmod(λ)) for the selected parameter combination. Step 108 is to select a different set of parameter values and repeat the calculation until all combinations are calculated. Step 110 is to provide estimates of parameter values based on a best fit between modeled R and measured R. Step 112 is to cache intermediate results that can be reused, such as eigenvalues and eigenvector matrices.
Although the various known approaches for reducing RCWA calculation time can be more or less effective, depending on circumstances, there remains an ongoing need to further reduce RCWA calculation time.